The present invention relates to a system and a computer implemented method for performing risk analysis of a portfolio.
The financial services industry, especially the financial risk management departments and the financial security pricing departments of insurance and re-insurance companies and banks, has established in the past tools and means for estimating their financial risk. Such risks can be associated with credit instruments and portfolios of credit instruments, such as bonds and loans. Such risks can also be associated with equity portfolios of various currencies or insurance and reinsurance liabilities.
These tools and means are based on models, based on which simulations are performed to generate possible valuation scenarios. These simulations generally use the Monte Carlo method or other appropriate methods. The models use probability distributions and are calibrated with historical data. Such historical data may be obtained from various sources, such as DataStream™.
These simulations are usually implemented in computer software as part of a financial services system and are run on computer hardware.
The input data for the simulations are risk factors, which are handled as random variables. Such risk factors can be equity indices, foreign exchange rates, interest rates, or insurance loss frequencies and severities. The result or output data of such simulations is at least one risk measure in the form of a numerical quantity or value. Usually, several risk measure values of different types can be obtained.
These risk measure values will be forwarded to an analyst or an actuary or an underwriter, i.e. a human representative of a financial services company. These risk measure values enable him to decide whether or not any actions should be taken to reduce the risk. Such actions can be changes in a credit or equities portfolio, or in a portfolio of insurance and reinsurance liabilities.
The risk measures usually consist of a variety of values, such as the maximum value obtained, the standard deviation of the simulation, a shortfall, usually the 99% shortfall, or a value-at-risk (VAR™). The VAR™ is the greatest possible loss that the company may expect in the portfolio in question with a certain given degree of probability during a certain future period of time. The full distribution itself can be the risk measure as well.
Typically a large number of risk factors have to be considered. Therefore, multidimensional probability distributions have to be used. As the risk measures are often determined at the tail of such distributions, a precise modelling of the tail dependency is important.
Furthermore, the dependency of the risk factors has to be considered. However, when using a linear correlation, the dependency is often not modelled adequately. One known solution to better model dependency is the use of copulas.
These copulas are well known in the state of the art. They are joint distribution functions of random vectors with standard uniform marginal distributions. They provide a way of understanding how marginal distributions of single risks are coupled together to form joint distributions of groups of risks.
Different kinds of copulas are known. Examples of closed form copulas are the Gumbel and the Clayton copula. Examples of implicit copulas, i.e. copulas for which no closed form exists, are the Gaussian copula and the t-copula.
It has become increasingly popular to model vectors of risk factor log returns with so-called meta-t distributions, i.e. distributions with a t-copula and arbitrary marginal distributions. The reason for this is the ability of the t-copula to model the extremal dependence of the risk factors and also the ease with which the parameters of the t-copula can be estimated from data. In Frey Rüdiger et al., “copulas and credit models” RISK, October 2001, p.p. 111-114, the use of such t-copulas for modelling credit portfolio losses is described. The disclosure thereof is herein implemented by reference.
We will recall therefore only the basic definitions and properties of t-distributions and t-copulas. For more on copulas in general, see NELSEN, R. (1999): An Introduction to copulas. Springer, New York, or EMBRECHTS, P., A. MCNEIL, AND D. STRATTON (2002): “Correlation and Dependence in Risk Management: Properties and Pitfalls,” in Risk Management: Value at Risk and Beyond, ed. By M. Dempster, pp. 176-223. Cambridge University Press, Cambridge.
Before describing the state of the art and the present invention in greater detail it is helpful to define the various variables and values. The following notation is used in the description of the prior art as well as of the invention:    d dimension, number of risk factors    Rd the d-dimensional usual real vector space    E(X) expected value of the random variable X    Var(X) variance of the random variable X    Cov (X,Y) covariance of the random variables X and Y    X,Y,Z random vectors    Cov(X) covariance matrix of X    ν number of degrees of freedom    Σ covariance matrix    Nd(0,Σ) d-dimensional Gaussian distribution with mean 0 and covariance Σ    φ univariate Gaussian distribution function    χν2 Chi Square distribution with degree of freedom ν    ρ correlation matrix    tν Student's t distribution function with degree of freedom ν    tν−1 Student's t quantile function    tν,ρd Student's t d-dimensional distribution function with correlation matrix ρ and degree of freedom ν    Γ usual gamma function    det A determinant of matrix A    Hk arbitrary univariate distribution function    U random variable uniformly distributed on [0,1]    τ(X,Y) Kendall's tau rank correlation for random variables X and Y    ak credit multi-factor model parameters    P└A┘ probability of occurrence of event A P└X≦x┘ probability that X is lower or equal than x    λk counterparty idiosyncratic parameter    Ek credit exposure on counterparty k    lk loss given default
Let Z˜Nd(0,Σ) and U (random variable uniformly distributed on [0,1]) be independent. Furthermore, G denotes the distribution function of √{square root over (ν|xν2)} and R=G−1(U).
Then the Rd—valued random vector Y given byY=(RZ1,RZ2,RZ3, . . . ,RZd)′  (1)has a centered t-distribution with ν degrees of freedom. Note that for ν>2,
      Cov    ⁡          (      Y      )        =            v              v        -        2              ⁢          Σ      .      By Sklar's Theorem, the copula of Y can be written asCν,ρt,(u)=tν,ρd(tν−1(u1), . . . ,tν−1(ud)),  (2)where ρi,j=Σij/√{square root over (ΣiiΣjj)} for i, jε{1, . . . , d} and where tν,ρd denotes the distribution function of √{square root over (ν)}Z/√{square root over (S)}, where S˜xν2 and Z˜Nd(0,ρ) are independent (i.e. the usual multivariate t distribution function) and tν denotes the marginal distribution function of tνρ,d (i.e. the usual univariate t distribution function). In the bivariate case the copula expression can be written as
                                          C                          v              ,              ρ                        t                    ⁡                      (                          u              ,              v                        )                          =                              ∫                          -              ∞                                      t              v                                                -                  1                                ⁢                                  (                  u                  )                                                              ⁢                                    ∫                              -                ∞                                            t                v                                                      -                    1                                    ⁢                                      (                    v                    )                                                                        ⁢                                          1                                  2                  ⁢                                                            π                      ⁡                                              (                                                  1                          -                                                      ρ                            12                            2                                                                          )                                                                                    1                      /                      2                                                                                  ⁢                                                {                                      1                    +                                                                                            s                          2                                                -                                                  2                          ⁢                                                      ρ                            12                                                    ⁢                          st                                                +                                                  t                          2                                                                                            v                        ⁡                                                  (                                                      1                            -                                                          ρ                              12                              2                                                                                )                                                                                                      }                                                                      -                                          (                                              v                        +                        2                                            )                                                        /                  2                                            ⁢                                                          ⁢                              ⅆ                s                            ⁢                                                          ⁢                                                ⅆ                  t                                .                                                                        (        3        )            
Note that ρ12 is simply the usual linear correlation coefficient of the corresponding bivariate tν-distribution if ν>2. The density function of the t-copula is given by
                                                        c                              v                ,                ρ                            t                        ⁡                          (                                                u                  1                                ,                …                ⁢                                                                  ,                                  u                  d                                            )                                =                                    1                                                det                  ⁢                                                                          ⁢                  ρ                                                      ⁢                                                                                Γ                    ⁡                                          (                                                                        v                          +                          d                                                2                                            )                                                        ⁢                                                            Γ                      ⁡                                              (                                                  v                          2                                                )                                                                                    d                      -                      1                                                                                                            Γ                    ⁡                                          (                                                                        v                          +                          1                                                2                                            )                                                        d                                            ·                                                                    ∏                                          k                      =                      1                                        d                                    ⁢                                                            (                                              1                        +                                                                              y                            k                            2                                                    v                                                                    )                                                                                      v                        +                        1                                            2                                                                                                            (                                          1                      +                                                                                                    y                            ′                                                    ⁢                                                      ρ                                                          -                              1                                                                                ⁢                          y                                                v                                                              )                                                                              v                      +                      d                                        2                                                                                      ,                            (        4        )            where yk=tν−1(uk).
Let H1, . . . , Hd be arbitrary continuous, strictly increasing distribution functions and let Y be given by (1) with Σ a linear correlation matrix. Thenx=(H1−1(tν(Y1)), . . . ,Hd−1(tν(Yd))′  (5)has a tν-copula and marginal distributions H1, . . . , Hd. The distribution of X is referred to as a meta-t distribution. Note that X has a t-distribution if and only if H1, . . . , Hd. are univariate tν-distribution functions.
The coefficient of tail dependence expresses the limiting conditional probability of joint quantile exceedences. The t-copula has upper and lower tail dependence with ( tν+1(x)=1−tν+1(x)):λ=2 tν+1(√{square root over (ν+1)}√{square root over (1−ρ12)}/√{square root over (1+ρ12)})<0,in contrast with the Gaussian copula which has λ=0. From the above expression it is also seen that the coefficient of tail dependence is increasing in ρ12 and, as one would expect since a t-distribution converges to a normal distribution as ν tends to infinity, decreasing in ν. Furthermore, the coefficient of upper (lower) tail dependence tends to zero as the number of degrees of freedom tends to infinity for ρ12<1.
The calibration of the copula parameters (ρ,ν) are typically done as follows:
(i) Kendall's tau ρ(Xi,Yj) is estimated for every pair of risk factor log returns. An estimate of the parameter ρ in (2) is obtained from the relation
                              τ          ⁡                      (                                          X                i                            ,                              Y                j                                      )                          =                              2            π                    ⁢                      arcsin            ⁡                          (                              ρ                ij                            )                                                          (        6        )            which holds for any distribution with strictly increasing marginal distribution functions and a copula of an elliptical distribution which has a density, i.e. essentially any meta-elliptical distribution one would consider in applications. Note that in high-dimensional applications an estimate of obtained from (6) may have to be modified to assure positive definiteness. This can be done by applying the so-called eigenvalue method, i.e. the negative eigenvalues are replaced by a small positive number. Other calibrations are possible too.
(ii) Transforming each log return observation Xi with its respective distribution function, e.g. gaussian N1(0,σi) yields, under the meta-t assumption, a sample from a t-copula with known ρ-parameter. Finally, the degrees of freedom parameter ν is estimated by standard maximum likelihood estimation using (4).
In step (ii) the empirical marginals or fitted distribution functions from a parametric family can be used.
The simulation from t-copula comprises the following steps:
(i) Draw independently a random variate Z from the d-dimensional normal distribution with zero mean, unit variances and linear correlation matrix ρ, and a random variate U from the uniform distribution on (0,1).
(ii) Obtain R by setting R≈Gν−1(U). By (1) we obtain a random variate Y from the t-distribution.
(iii) Finally,(tν(Y1), . . . ,tν(Yd))′is a random variate from the t-copula
This meta-t assumption makes sense for risk factors of similar type, e.g. foreign exchange rates. However, it was found that it does not accurately describe the dependence structure for a set of risk factor log returns where the risk factors are of very different type, for example a mixture of stock indices, foreign exchange rates and interest rates.
It is a general problem of such models, that the number of available historical data is quite small, so that at least the tail dependency can hardly be modelled. Similar problems are also known in other fields, for example in the combination reinsurance portfolios, the reliability of industrial complexes or in the weather forecast.